IEEE Power Electronics Magazine - March 2020 - 44

Table 5. Available well-behaved measurements for the proposed model, extended to four windings
in Figure 10. While some of the measurements require short circuit terminations at ports,
none of the tests reported here require current sensing.
Apply Condition to Winding
1

2

3

4

Measure

m1

v1

Open circuit

Open circuit

Open circuit

Z1

m2

v1

Open circuit

Open circuit

Open circuit

v 2 /v 1

= n2

m3

v1

Open circuit

Open circuit

Open circuit

v 3 /v 1

= n3

m4

v1

Open circuit

Open circuit

Open circuit

v 4 /v 1

= n4

m5

v1

Open circuit

Open circuit

Short circuit

v 2 /v 1

= (L 4 + L 5)/(L 1 + L 4 + L 5) # n 2

m6

v1

Open circuit

Open circuit

Short circuit

v 3 /v 1

= L 4 /(L 1 + L 4 + L 5) # n 3

m7

v1

Open circuit

Short circuit

Open circuit

v 2 /v 1

= (L 3 + L 5)/(L 1 + L 3 + L 5 + L 6) # n 2

m8

v1

Open circuit

Short circuit

Open circuit

v 4 /v 1

= (L 3 + L 6)/(L 1 + L 3 + L 5 + L 6) # n 4

= j~L m

m9

v1

Short circuit

Open circuit

Open circuit

v 3 /v 1

= L 2 /(L 1 + L 2 + L 6) # n 3

m 10

v1

Short circuit

Open circuit

Open circuit

v 4 /v 1

= (L 2 + L 6)/(L 1 + L 2 + L 6) # n 4

m 11

Short circuit

v2

Open circuit

Open circuit

Z2

= j~ (L 1 + L 2 + L 6) # n 22

m 12

Short circuit

Open circuit

v3

Open circuit

Z3

= j~ (L 1 + L 3 + L 5 + L 6) # n 32

m 13

Short circuit

Open circuit

Open circuit

v4

Z4

= j~ (L 1 + L 4 + L 5) # n 24

m 14

Short circuit

v2

Open circuit

Open circuit

v 3 /v 2

= (L 1 + L 6)/(L 1 + L 2 + L 6) # n 3 /n 2

m 15

Short circuit

v2

Open circuit

Open circuit

v 4 /v 2

= L 1 /(L 1 + L 2 + L 6) # n 4 /n 2

m 16

Short circuit

Open circuit

v3

Open circuit

v 2 /v 3

= (L 1 + L 6)/(L 1 + L 3 + L 5 + L 6) # n 2 /n 3

m 17

Short circuit

Open circuit

v3

Open circuit

v 4 /v 3

= (L 1 + L 5)/(L 1 + L 3 + L 5 + L 6) # n 4 /n 3

m 18

Short circuit

Open circuit

Open circuit

v4

v 2 /v 4

= L 1 /(L 1 + L 4 + L 5) # n 2 /n 4

m 19

Short circuit

Open circuit

Open circuit

v4

v 3 /v 4

= (L 1 + L 5)/(L 1 + L 4 + L 5) # n 3 /n 4

Table 6. Well-behaved measurements for the
proposed four-winding model can be used to
derive seven of the 10 model parameters without
subtraction, an improvement over four of 10 for
the four-winding extended cantilever model.

44	

Lm

m1

n2

m2

n3

m3

n4

m4

L1

m 18 m 13 /(m 2 m 4) or m 15 m 11 /(m 2 m 4)

L2

m 9 m 11 /(m 3 m 22)

L3

-

L4

m 6 m 13 /(m 3 m 24)

L5

-

L6

-

IEEE POWER ELECTRONICS MAGAZINE	

z	March 2020

calculations in Table 6. From Table 6, it is clear that the proposed model for four-winding structures can yield seven
of 10 parameters with fully well-behaved measurements
and calculations (more than the four of 10 such parameters of the extended cantilever model). The remaining
three parameters may be obtained by less desirable means
(subtraction, current sensing, and others) where sensitivity allows, and there are several ways to obtain each of the
remaining parameters.

Conclusions
As designers strive to best take advantage of good
switching devices and best grapple with the magnetic
bottleneck, the solution often involves more complex
magnetic structures. Modeling such structures with mathematics, necessary-and-sufficient circuit models, and
physical circuit models is an essential element in understanding their complicated behavior. Our brief review has
covered some of the primary modeling approaches, their
IEEE Power Electronics Magazine - March 2020

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